A005565 Number of walks on square lattice. (Formerly M5087)

20, 75, 189, 392, 720, 1215, 1925, 2904, 4212, 5915, 8085, 10800, 14144, 18207, 23085, 28880, 35700, 43659, 52877, 63480, 75600, 89375, 104949, 122472, 142100, 163995, 188325, 215264, 244992, 277695, 313565, 352800, 395604, 442187, 492765, 547560, 606800

Offset

0,1

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

R. K. Guy, Letter to N. J. A. Sloane, May 1990

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = 1/4*(n^4+14n^3+69n^2+136n+80). G.f.: (20-25x+14x^2-3x^3)/(1-x)^5. - Ralf Stephan, Apr 23 2004

a(n) = binomial(n+4,2)^2 - binomial(n+4,1)^2. - Gary Detlefs, Nov 22 2011

Using two consecutive triangular numbers t(n) and t(n+1), starting at n=3, compute the determinant of a 2 X 2 matrix with the first row t(n), t(n+1) and the second row t(n+1), 2*t(n+1). This gives (n+1)^2*(n-2)*(n+2)/4 = a(n-3). - J. M. Bergot, May 17 2012

Maple

seq(add (k^3-n^2, k =0..n), n=4..28 ); # Zerinvary Lajos, Aug 26 2007
A005565:=(-20+25*z-14*z**2+3*z**3)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

CoefficientList[Series[(20-25x+14x^2-3x^3)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 24 2012 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {20, 75, 189, 392, 720}, 40] (* Harvey P. Dale, Dec 04 2020 *)

Prog

(PARI) a(n)=(n^4+14*n^3+69*n^2+136*n)/4+20 \\ Charles R Greathouse IV, Nov 22 2011
(Magma) [1/4*(n^4+14*n^3+69*n^2+136*n+80): n in [0..40]]; // Vincenzo Librandi, May 24 2012

Crossrefs

Sequence in context: A237617 A000529 A238027 * A320484 A066126 A228025

Adjacent sequences: A005562 A005563 A005564 * A005566 A005567 A005568

Keyword

nonn,walk,easy

Author

N. J. A. Sloane

Status

approved